Let \(X^n\) be a nonsingular hypersurface of degree \(d\ge 2\) in the projective space \(\mathbb {P}^{n+1}\) defined over a finite field \(\mathbb {F}_q\) of q elements. We prove a Homma–Kim conjecture on a upper bound about the number of \(\mathbb {F}_q\)-points of \(X^n\) for \(n=3\), and for any odd integer \(n\ge 5\) and \(d\le q\).

In this paper, we establish a local regularity result for \(W^{2,p}_{{\mathrm {loc}}}\) solutions to complex degenerate nonlinear elliptic equations \(F(D^2_{\mathbb {C}}u)=f\) when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation.

In this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity \(|u|^p\) or nonlinearity of derivative type \(|u_t|^p\), in any space dimension \(n\geqslant 1\), for supercritical powers \(p>{\bar{p}}\). The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive \(L^r-L^q\)

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group \(S_d\). We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces

In Chen and Wang (Calc Var Part Differ Equ 56:1–26, 2017), we show that, if \(\epsilon >0\) is small enough, then there exists a sequence of semiclassical states of higher topological type localized at a local minimum set of the potential V for the semiclassical nonlinear Schrödinger equation $$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1\left( {\mathbb{R}}^N\right) . \end{aligned}$$

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group \({\mathbb {H}}^n\), \(n\in {\mathbb {N}}\). For \(1\leqslant k\leqslant n\), we show that every intrinsic L-Lipschitz graph over a subset of a k-dimensional horizontal subgroup \({\mathbb {V}}\) of \({\mathbb {H}}^n\) can be extended to an intrinsic \(L'\)-Lipschitz

Given a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties

We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur

We consider a uniform r-bundle E on a complex rational homogeneous space X and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or unstable with respect to some numerical class of a line. So we partially answer a problem posted by Muñoz et al. (Eur

We prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS, thus extending what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result, it is very simple to deduce a result of exponential tightness for Gaussian probabilities.

In this paper, we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on \(\mathrm {SU}(n,1)/\mathrm {S}(\mathrm {U}(n)\mathrm {U}(1))\). We use this characterisation

For \(p \in (1,N)\) and \(\Omega \subseteq {\mathbb {R}}^N\) open, the Beppo-Levi space \({\mathcal {D}}^{1,p}_0(\Omega )\) is the completion of \(C_c^{\infty }(\Omega )\) with respect to the norm \(\left[ \int _{\Omega }|\nabla u|^p \ dx \right] ^ \frac{1}{p}.\) Using the p-capacity, we define a norm and then identify the Banach function space \({\mathcal {H}}(\Omega )\) with the set of all g in \(L^1_{loc}(\Omega

We establish weighted \(L^p\)-Fourier extension estimates for \(O(N-k) \times O(k)\)-invariant functions defined on the unit sphere \({\mathbb {S}}^{N-1}\), allowing for exponents p below the Stein–Tomas critical exponent \(\frac{2(N+1)}{N-1}\). Moreover, in the more general setting of an arbitrary closed subgroup \(G \subset O(N)\) and G-invariant functions, we study the implications of weighted Fourier

In this paper, we consider a pseudo-parabolic equation, which was studied extensively in recent years. We generalize and extend the existing results in the following three aspects. First, we consider the vacuum isolating phenomenon with the initial energy \(J(u_0)\) satisfying \(J(u_0)\le 0\) and \(0

In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which

Let \(1 \le p \le \infty \). A Banach lattice X is said to be p-disjointly homogeneous or \((p-DH)\) (resp. restricted \((p-DH)\)) if every normalized disjoint sequence in X (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in X to the unit vector basis of \(\ell _p\). We revisit DH-properties of Orlicz spaces and refine some previous

We describe the structure of the projective cover of a module \(M_R\) over a local ring R and its relation with minimal sets of generators of \(M_R\). The behaviour of local right perfect rings is completely different from the behaviour of local rings that are not right perfect.

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local

An octonionic contact (OC) manifold is always spherical. We construct the OC Yamabe operator on an OC manifold and prove its transformation formula under conformal OC transformations. An OC manifold is scalar positive, negative or vanishing if and only if its OC Yamabe invariant is positive, negative or zero, respectively. On a scalar positive OC manifold, we can construct the Green function of the

The paper concerns the study of the Cauchy–Dirichlet problem for a class of hyperbolic second-order operators with double characteristics in presence of transition in a domain of \({\mathbb {R}}^3\). Firstly, we establish some a priori local and global estimates. Then, we obtain some existence results.

In the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures \(f(k_1,\ldots ,k_{n-1})\) satisfying suitable conditions. In this paper, we give sharp quantitative estimates of proximity to a single sphere for Alexandrov

We classify complex surfaces \((M,\,J)\) admitting Engel structures \({\mathcal {D}}\) which are complex line bundles. Namely, we prove that this happens if and only if \((M,\,J)\) has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges [7]. We also study associated Engel defining forms and define a unique splitting of TM associated with \({\mathcal

Given an arbitrary \(C^\infty \) Riemannian manifold \(M^n\), we consider the problem of introducing and constructing minimal hypersurfaces in \(M\times \mathbb {R}\) which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space \(\mathbb {R}^3=\mathbb {R} ^2\times \mathbb {R}\). Such hypersurfaces are defined by imposing conditions on their height functions

In this paper we prove higher order Poincaré inequalities involving radial derivatives namely, $$\begin{aligned} \int _{\mathbb {H}^{N}} |\nabla _{r,\mathbb {H}^{N}}^{k} u|^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ge \bigg (\frac{N-1}{2}\bigg )^{2(k-l)} \int _{\mathbb {H}^{N}} |\nabla _{r,\mathbb {H}^{N}}^{l} u|^2 \, \mathrm{d}v_{\mathbb {H}^{N}} \ \ \text { for all } u\in H^k(\mathbb {H}^{N}), \end{aligned}$$

In the present article we study a minimization problem in \(\mathbb R^N\) involving the perimeter of the positivity set of the solution u and the integral of \(|\nabla u|^{p(x)}\). Here p(x) is a Lipschitz continuous function such that \(10\}\) is a \(C^\infty \) surface.

We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of \({\mathbb {C}}^{d}\) admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to \({\mathbb {C}}\times ({\mathbb {C}}^{*})^{d-1}\) and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or

We study a free transmission problem in which solution minimizes a functional with different definitions in positive and negative phases. We prove some asymptotic regularity results when the jumps of the diffusion coefficients get smaller along the free boundary. At last, we prove a measure-theoretic result related to the free boundary.

The Debarre-de Jong conjecture predicts that the Fano variety of lines on a smooth Fano hypersurface in \(\mathbb {P}^n\) is always of the expected dimension. We generalize this conjecture to the case of smooth Fano complete intersections and prove that for a smooth Fano complete intersection \(X\subset \mathbb {P}^n\) of hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on X

We study equilibrium configurations for the Euler–Plateau energy with elastic modulus, which couples an energy functional of Euler–Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is shown that the potential minimizers of this energy are highly dependent on the choice of physical rigidity parameters, and that the area of critical surfaces can

Let f be analytic in the unit disk \({\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}\), and \({{\mathcal {S}}}\) be the subclass of normalized univalent functions given by \(f(z)=z+\sum _{n=2}^{\infty }a_n z^n\) for \(z\in {\mathbb {D}}\). We give sharp bounds for the modulus of the second Hankel determinant \( H_2(2)(f)=a_2a_4-a_3^2\) for the subclass \( {\mathcal F_{O}}(\lambda ,\beta )\) of strongly

Let K be affine, that is, \(K=\{z=(x,y)\in {\mathbb {R}}^{n+m}: y_{1}=\cdots =y_{m}=0\}\). We compute the sharp constant of Hardy inequality related to the distance d(z, K) for polyharmonic operator. Moreover, we show that there exists a constant \(C>0\) such that for each \(u\in C^{\infty }_{0}({\mathbb {R}}^{n+m}\setminus K)\), there holds $$\begin{aligned} \int _{{\mathbb {R}}^{n+m}}|\nabla ^{k}

In this paper we obtain symmetry and monotonicity results for positive solutions to some p-Laplacian cooperative systems in bounded domains involving first-order terms and under zero Dirichlet boundary condition.

In proving his theorem on global maximizers for the sphere adjoint Fourier restriction inequality, D. Foschi studied some quadratic functional and determined its optimal bound. In this note we extend to higher-dimensional spheres his results on this quadratic functional.

This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system and the global hypoellipticity of its normal form. It is proved that an obstruction of number-theoretical nature appears as a necessary condition to the global hypoellipticity

In this paper, we are concerned with the problem $$\begin{aligned} -\text{ div } \left( \displaystyle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$ where \(\Omega \subset {\mathbb {R}}^{2}\) is a bounded smooth domain and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a superlinear continuous function with

We know that there are infinitely many sprays of scalar curvature which cannot be induced by any (not necessary positive definite) Finsler metric. In this paper, we show that every spray of scalar curvature satisfies the following: the rate of change of the Douglas curvature along a geodesic is tangent to the geodesic, generalizing Sakaguchi result previously known only when the spray is induced by

Using a result of Demailly and Pham on log canonical thresholds, we give an upper bound for polar invariants from a question of Teissier on hypersurface singularities. This provides a weaker alternative upper bound compared to the one conjectured by Teissier.

In this article, we show the following result: If C is an n-dimensional convex and compact subset, \(f:C\rightarrow [0,\infty )\) is concave, and \(\phi :[0,\infty )\rightarrow [0,\infty )\) is a convex function with \(\phi (0)=0\), we then characterize the class of sets and concave functions that attain the supremum $$\begin{aligned} \sup _{C,f}\int _C\phi (f(x)){\text {d}}x, \end{aligned}$$ where

In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher-order Lorentz–Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in Nguyen VH (J Math Anal Appl, 490(1):124197, 2020) to the higher-order derivatives and seem to be new in the context of the Lorentz–Sobolev spaces defined in the hyperbolic spaces.

We analytically characterize equilateral triangles. Our characterization includes the classically well-known characterizations: for a given triangle in the Euclidean plane, if the centroid and incenter of the triangle coincide, then the triangle must be equilateral; for a given triangle in the Euclidean plane, if the centroid and circumcenter of the triangle coincide, then the triangle must be equilateral

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.

We consider an energy functional combining the square of the local oscillation of a one-dimensional function with a double-well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties. In addition, we investigate the main properties of

We study left-invariant generalized Kähler structures on almost abelian Lie groups, i.e., on solvable Lie groups with a codimension-one abelian normal subgroup. In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is \(\partial {{\overline{\partial

We analyze the steady motion of a viscous incompressible fluid in a three-dimensional channel containing an obstacle through the Navier–Stokes equations with mixed boundary conditions: the inflow is given by a fairly general datum and the flow is assumed to satisfy a constant traction boundary condition on the outlet, together with the standard no-slip assumption on the obstacle and on the remaining

Motivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled

The Olver–Benney equation is a nonlinear fifth-order equation, which describes the interaction effects between short and long waves. In this paper, we prove the global existence of solutions of the Cauchy problem associated with this equation.

We prove that the Euclidean ball can be realized as a Fatou component of a holomorphic automorphism of \(\mathbb{C}^m\), in particular as the escaping and the oscillating wandering domain. Moreover, the same is true for a large class of bounded domains, namely for all bounded simply connected regular open sets \(\Omega \subset \mathbb{C}^m\) whose closure is polynomially convex. Our result gives in

We consider the complex hyperbolic quadric \({Q^*}^n\) as a complex hypersurface of complex anti-de Sitter space. Shape operators of this submanifold give rise to a family of local almost product structures on \({Q^*}^n\), which are then used to define local angle functions on any Lagrangian submanifold of \({Q^*}^n\). We prove that a Lagrangian immersion into \({Q^*}^n\) can be seen as the Gauss map

Given any diagonal cyclic subgroup \(\Lambda \subset \text {GL}(n+1,k)\) of order d, let \(I_d\subset k[x_0,\ldots , x_n]\) be the ideal generated by all monomials \(\{m_{1},\ldots , m_{r}\}\) of degree d which are invariants of \(\Lambda\). \(I_d\) is a monomial Togliatti system, provided \(r \le \left( {\begin{array}{c}d+n-1\\ n-1\end{array}}\right)\), and in this case the projective toric variety

In this note, we formulate an explicit formula for the completed zeta function of a function field K of genus g over a finite field \(\mathbb {F}_q\), analogous to the Weil explicit formula. Then we give an arithmetic formula for the nth Li coefficient \(\lambda _{K}(n)\) for the function field K. Furthermore, as an application, we give some formulas for the Euler–Stieltjes constants \(\gamma _{K}\)

We study the functional calculus associated with a hypoelliptic left-invariant differential operator \(\mathcal {L}\) on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator \(\mathcal {L}_0\) on the ‘local’ contraction \(G_0\) of G, as well as of the corresponding Rockland operator \(\mathcal {L}_\infty \) on the ‘global’ contraction \(G_\infty

We use Perron method to prove the existence of ancient solutions of exterior problem for a kind of parabolic Monge–Ampère equation \(-\,u_t\det D^2u=f\) with prescribed asymptotic behavior at infinity outside some certain bowl-shaped domain in the lower half space for \(n\ge 3\), where f is a perturbation of 1 at infinity. We raise this problem for the first time and construct a new subsolution to

It is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the

The regularity for the 3-D nematic liquid crystal equations is considered in this paper, it is proved that the Leray–Hopf weak solutions (u, d) is in fact smooth, if the velocity field \(u\in L^\infty (0,T;L^{3,\infty }_x(\mathbb {R}^3))\) satisfies some addition local small condition $$\begin{aligned} r^{-3}\left| \left\{ x\in B_r(x_0): |u(x,t_0)|>\varepsilon r^{-1}\right\} \right| \le \varepsilon

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kähler, we solve the Hermitian–Einstein equation on analytically stable Higgs bundles.

In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold M for which there exists a bounded vector field X such that \(\langle \nabla f,X\rangle \ge 0\) on M and \(\rm {div} X\ge af\) outside a suitable compact subset of M, for some constant \(a>0\), under the assumption that M has either polynomial or exponential volume growth. We then

Given a two-dimensional mapping U whose components solve a divergence structure elliptic equation, we give necessary and sufficient conditions on the boundary so that U is a global diffeomorphism.

We consider the following prescribed curvature problem involving polyharmonic operator: $$\begin{aligned} D_mu=Q(|y'|,y'')u^{m^*-1}, \;u>0, \; u \in {\mathcal {H}}^{m}({\mathbb {S}}^{N}), \end{aligned}$$ where \(m^*=\frac{2N}{N-2m},\; N\ge 4m+1\), \(m \in {\mathbb {N}}_+\), \((y',y'') \in {\mathbb {R}}^{2} \times {\mathbb {R}}^{N-2}\), and \(Q(|y'|,y'')\) is a bounded nonnegative function in \({\mathbb

Starting from the pioneering work by Meeks, complete nonorientable minimal surfaces with finite total curvature have been studied by many researchers. However, it seems that there are no known examples all of whose ends are embedded except for Kusner’s flat-ended N-noids. In this paper, we show the existence of a 1-parameter family of complete \({\mathbf{Z}}_N\)-invariant conformal minimal immersions